Remarks on punctual local connectedness

Peter Johnstone

We study the condition, on a connected and locally connected
geometric morphism $p : \cal E \to \cal S$, that the
canonical natural transformation $p_*\to p_!$
should be (pointwise) epimorphic - a condition which F.W. Lawvere
called the `Nullstellensatz', but which we prefer to
call `punctual local connectedness'. We show that this condition
implies that $p_!$ preserves finite products, and that, for
bounded morphisms between toposes with natural number objects,
it is equivalent to being both local and hyperconnected.